Problem 42 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\lim _{n \rightarrow \infty} \frac{1}{n}\left(1+e^{1 / n}+e^{2 / n}+\ldots+e^{\frac{n-1}{n}}\right)\) is equal to (A) \(e\) (B) \(-e\) (C) \(e-1\) (D) \(1-e\)

Step-by-Step Solution

Answer

(C) \(e-1\)

Step 1: Understand the Limit Expression

The given limit is \( \lim _{n \rightarrow \infty} \frac{1}{n}\left(1+e^{1 / n}+e^{2 / n}+\ldots+e^{\frac{n-1}{n}}\right) \). This expression involves a sum of exponential terms divided by \( n \), suggesting a connection to the Riemann sum for integration.

Step 2: Recognize Riemann Sum Structure

Notice that the sum inside the limit resembles a Riemann sum. The Riemann sum \( \sum_{k=0}^{n-1} e^{k/n} \cdot \frac{1}{n} \) approximates the integral \( \int_0^1 e^x \, dx \) as \( n \to \infty \). Our expression has such a structure.

Step 3: Convert to Riemann Integral

The limit expression \( \lim _{n \rightarrow \infty} \frac{1}{n}\sum_{k=0}^{n-1} e^{k/n} \) is equivalent to the integral \( \int_0^1 e^x \, dx \). The function \( e^{x} \) will be integrated over \([0, 1]\).

Step 4: Solve the Integral

Compute the integral \( \int_0^1 e^x \, dx \). The antiderivative of \( e^x \) is \( e^x \). Evaluating this from 0 to 1, we get \( e^1 - e^0 = e - 1 \).

Step 5: Conclude the Solution

Given that the limit converts to the integral \( \int_0^1 e^x \, dx = e - 1 \), the correct answer is \( e - 1 \). Therefore, the answer is (C).

Key Concepts

Riemann Sums

Riemann sums are a fundamental concept in calculus that allow us to approximate the area under a curve. Imagine you want to find the area under a curve from one point to another. A Riemann sum does this by breaking the area into rectangles, and the sum of these rectangle areas gives an approximation of the total area. You can improve accuracy by increasing the number of rectangles. Here's how it works:

• Choose a number of rectangles, say "n".
• The width of each rectangle is determined by dividing the total interval by "n".
• The height is found by evaluating the function at specific points within each sub-interval.

As the number of rectangles increases (or as "n" approaches infinity), the sum of these approximations becomes more accurate, essentially mirroring the calculation of a definite integral. This limit of the Riemann sum is an integral, which accurately measures the area under a curve.

Exponential Functions

Exponential functions involve base numbers raised to a variable exponent and are characterized by rapid growth. The function you often see is of the form \(f(x) = e^x\), where "e" is a mathematical constant approximately equal to 2.71828. This constant is unique because it arises naturally in mathematics, particularly in growth and decay problems.Exponential growth can be seen in various natural processes:

• Population growth where new individuals add based on a proportion of the existing population.
• Compound interest in finance, where the amount grows based on the initial amount plus the interest over time.

Exponential functions are continuous and smooth, making them ideal for modeling natural phenomena, and they have unique properties. For instance, the rate of change of the function \(e^x\) is equal to the function itself, implying that as you increase "x", \(e^x\) translates into very rapid growth.

Definite Integrals

Definite integrals represent the area under a curve between two points, with precise limits. Unlike an indefinite integral, which represents a family of functions and includes an arbitrary constant, a definite integral has fixed limits and results in a specific numerical value.Let's see how definite integrals work:

• The integral sign \(\int\) indicates integration, along with limits of integration (like between 0 and 1).
• Definite integrals are dependent on the Fundamental Theorem of Calculus, which connects differentiation and integration.

To solve a definite integral, you'd find the antiderivative of the function (the parent function before differentiation) and evaluate it at the upper and lower limits. Using these limits, you subtract the value of the antiderivative at the lower limit from its value at the upper limit to find the total area under the curve within those limits.In our problem, the definite integral \(\int_0^1 e^x \, dx\) represents finding the area under the exponential function \(e^x\) between 0 and 1.